Power Systems Stability Analysis Based on Classical Techniques in Work

Abstract

This chapter presents a linearized Phillips–Heffron model of a parallel AC/DC power system in order to studying power system stability. In addition, a supplementary controller for a modeling back-to-back voltage source converter (BtB VSC) HVDC to damp low-frequency oscillations in a weakly connected system is proposed. Also, input controllability measurement for BtB VSC HVDC is investigated using relative gain array (RGA), singular value decomposition (SVD) and damping function (DF) and a supplementary controller is designed based on phase compensating method. In addition, a supplementary controller for a novel modeling VSC HVDC to damp low-frequency oscillations in a weakly connected system is proposed. The potential of the VSC HVDC supplementary controllers to enhance the dynamic stability is evaluated by measuring the electromechanical controllability through SVD analysis. The presented control scheme not only performs damping oscillations but also the voltage and power flow control can be achieved. Simulation results obtained by MATLAB verify the effectiveness of the VSC HVDC and its control strategy for enhancing dynamical stability. Moreover, a linearized model of a power system installed with a UPFC has been presented. UPFC has four control loops that, by adding an extra signal to one of them, increases dynamic stability and load angle oscillations are damped. To increase stability, a novel online adaptive controllers have been used analytically to identify power system parameters. Suitable operation of adaptive controllers to decrease rotor speed oscillations against input mechanical torque disturbances is confirmed by the simulation results.

Appendices

Appendices

1.1 Appendix 1. Parameters of Test System (pu) for BtB VSC HVDC Network

Machine and Exciter: $\begin{gathered} x_{d} = 1,\;x_{q} = 0.6,\;x_{d}^{\prime} = 0.3,\;D = 0,\;M = 8, \hfill \\ T_{\rm do}^{\prime} = 5.044,\;f = 60,\;v_{\rm ref} = 1,\;K_{A} = 120,\;T_{A} = 0.015 \hfill \\ \end{gathered}$

Transmission line and transformer reactance: $X_{\rm tl} = 0.15,\;X_{\rm lb} = 0.6,\;X_{\rm sp} = X_{s} = 0.15$

BtB VSC HVDC: $V_{\rm dc} = 3,\;C_{\rm dc} = 1$.

Neural controller: two multilayer feed forward neural network with activation function: $a\,\tanh (\mathrm{bx})$.

Hidden and output layer for identifier includes 4 and 1 neuron, respectively with $a = b = 1.2,\;Eta = 0.4.$

Hidden and output layer for controller includes 3 and 1 neuron, respectively with $a = 17,b = 0.65,\;Eta = 0.22$.

1.2 Appendix 2. Parameters of Test System (pu) for VSC HVDC Network

Machine and Exciter: $\begin{gathered} x_{d} = 1,\;x_{q} = 0.6,\;x_{d}^{\prime} = 0.3,\;D = 0,\;M = 8, \hfill \\ T_{\rm do}^{\prime} = 5.044,\;f = 60,\;v_{\rm ref} = 1,\;K_{A} = 120,\;T_{A} = 0.015 \hfill \\ \end{gathered}$

Transmission line and transformer reactance: $x_{t} = 0.1,\;x_{l} = 1,\;x_{r} = x_{i} = 0.15$

VSC HVDC: $V_{\rm dcr} = 2,V_{\rm dci} = 1.95,\;C_{r} = C_{i} = 2,\;L_{1} = L_{2} = 0.09,\;C = 0.09$

1.3 Appendix 3. Constant Coefficients of the Linearised Dynamic Model of Eq. (59)

$$\begin{aligned} & Z = 1 + \frac{{x_{l}}}{{x_{r}}},\;A = x_{t} + x_{l}+ \frac{{x_{t}}}{{x_{r}}},\;[A] = A + Zx_{d}^{\prime},\;[B] = A +Zx_{q}\\ & C_{1} = \frac{{V_{b} \cos (\delta)}}{[B]},\;C_{2} = -\frac{{x_{l} M_{r} V_{\rm dc} \sin (\varphi_{r})}}{{2x_{r}[B]}},\;C_{3} = \frac{{x_{l} V_{\rm dc} \cos (\varphi_{r})}}{{2x_{r}[B]}},\;C_{4} = \frac{{x_{l} M_{r} \cos (\varphi_{r})}}{{2x_{r}[B]}}\\ & C_{5} = \frac{Z}{A},\;C_{6} = \frac{{V_{b} \sin(\delta)}}{[A]},\;C_{7} = - \frac{{x_{l} M_{r} V_{\rm dc} \cos(\varphi_{r})}}{{2x_{r} [A]}},\;C8 = - \frac{{x_{l} V_{\rm dc} \sin(\varphi_{r})}}{{2x_{r} [A]}},\\ & C_{9} = - \frac{{x_{l} M_{r}V_{\rm dc} \cos (\varphi_{r})}}{{2X_{r} [A]}} \\ & C_{a} = (x_{q} -x_{d}^{\prime})I_{t},\;C_{b} = E_{q}^{\prime} + (x_{q} -x_{d}^{\prime}),\;K_{1} = C_{b} C_{1} + C_{a} C_{6},\\ & K_{2} =I_{t} (1 + (x_{q} - x_{d}^{\prime})C_{5})\\ & K_{\rm pdcr} = C_{b}C_{4} + C_{a} C_{9},\;K_{{p{M_r}}} = C_{b} C_{3} + C_{a}C_{8},\;K_{p\varphi r} = C_{b} C_{2} + C_{a} C_{7} \\ & K_{3} = 1 +JC_{5},\;K_{4} = JC_{6},\\ & K_{q\varphi r} = JC_{7},\;K_{{q{M_r}}}= JC_{8},\;K_{\rm qdcr} = JC_{9}\\ & K_{5} = L(V_{\rm td} x_{q}C_{1} - V_{\rm tq} x_{d}^{\prime} C_{6}),\;K_{6} = LV_{\rm tq} (1 -x_{d}^{\prime} C_{5})\\ & K_{{{V_{\rm dc}}r}} = L(V_{\rm td} x_{q}C_{4} - V_{\rm tq} x_{d}^{\prime} C_{9}),K_{{V{M_r}}} = L(V_{\rm td}x_{q} C_{3} - V_{\rm tq} x_{d}^{\prime} C_{8}),K_{V\varphi r} =L(V_{\rm td} x_{q} C_{2} - V_{\rm tq} x_{d}^{\prime} C_{7})\\ &x_{d}- x_{d}^{\prime} = J,\;L = \frac{1}{{V_{t}}},\;E =\frac{{x_{d}^{\prime} + x_{t}}}{{x_{r}}},\;F = \frac{{x_{q} +x_{t}}}{{x_{r}}}\\ & C_{10} = EC_{5} - \frac{1}{{x_{r}}},\;C_{11} =EC_{6},\;C_{12} = EC_{7} - \frac{{M_{r}}}{{2x_{r}}}V_{\rm dcr} \sin(\varphi_{r}),\\ & C_{13} = \frac{1}{{2x_{r}}}M_{r} \cos(\varphi_{r}) + EC_{8} \\& C_{14} = \frac{1}{{2x_{r}}}\cos(\varphi_{r}) + EC_{9},\;C_{15} = FC_{1},\;C_{16} =\frac{1}{{2x_{r}}}V_{\rm dcr} \sin (\varphi_{r}) + FC_{2},\\ &C_{17} = - \frac{1}{{2x_{r}}}M_{r} \cos (\varphi_{r}) + FC_{4} \\ &C_{18} = FC_{3} - \frac{1}{{2x_{r}}}V_{\rm dcr} \cos(\varphi_{r}),\;C_{19} = \frac{1}{{x_{i}}}V_{\rm bd},\;C_{20} =\frac{1}{{2x_{i}}}M_{i} \sin (\varphi_{i}),\\ & C_{21} =\frac{1}{{2x_{i}}}V_{\rm dcr} \sin (\varphi_{i}) \\& C_{22} =\frac{1}{{2x_{i}}}M_{i} V_{\rm dci} \cos (\varphi_{i}),\;C_{23} =\frac{1}{{x_{i}}}V_{\rm bq},\;C_{24} = - \frac{1}{{2x_{i}}}M_{i}\cos (\varphi_{i}),\\ & C_{25} = - \frac{1}{{2x_{i}}}V_{\rm dci}\cos (\varphi_{i}),\;C_{26} = \frac{1}{{2x_{i}}}V_{\rm dci} \sin(\varphi_{i}) \\ & f_{1} = [0.5\cos (\varphi_{i})I_{\rm id} +0.5\sin (\varphi_{i})I_{\rm iq}],\;f_{2} = - [- 0.5\sin(\varphi_{i})I_{\rm id} + 0.5\cos (\varphi_{i})I_{\rm iq}] \\& f_{3}= - 0.5M_{i} \cos (\varphi_{i}),\;f_{4} = - 0.5M_{i} \sin(\varphi_{i}),\;f_{5} = - [0.5\cos (\varphi_{r})I_{\rm rd} + 0.5\sin(\varphi_{i})I_{\rm rq}]\\& f_{6} = - [- 0.5\sin (\varphi_{i})I_{\rm rd} + 0.5\cos(\varphi_{i})I_{\rm rq}],\;f_{7} = - 0.5M_{r} \cos(\varphi_{r}),\;f_{4} = - 0.5M_{r} \sin (\varphi_{r}) \\ & C_{27} =f_{3} C_{19} + f_{4} C_{23},\;C_{28} = f_{3} C_{20} + f_{4}C_{24},\;C_{29} = f_{1} + f_{3} C_{21} + f_{4} C_{25},\\ & C_{30} =f_{2} + f_{3} C_{22} + f_{4} C_{26}\\ & C_{31} = f_{7} C_{11} +f_{8} C_{15},\;C_{32} = f_{7} C_{10},\;C_{33} = f_{7} C_{14} + f_{8}C_{17},\;C_{34} = f_{5} + f_{7} C_{13} + f_{8} C_{18},\\ & C_{35} =f_{6} + f_{7} C_{12} + f_{8} C_{16}\end{aligned}$$

1.4 Appendix 4. The Test System Parameters for UPFC HVDC Network

Generator: $M = 2H = 8. 0 {\text{ MJ/MVA}}$, $D = 0. 0$, $T_{\rm do}^{\prime} = 5. 0 4 4 {\text{ s}}$, $x_{d} = 1. 0 {\text{ pu}}$, $x_{q} = 0. 6 {\text{ pu}}$, $x_{d}^{\prime} = 0. 3 {\text{ pu}}$

Excitation System: $K_{a} = 1 0 0$, $T_{a} = 0. 0 1 {\text{ s}}$

Transformer: $X_{\rm tE} = 0. 1 {\text{ pu}}$, $X_{E} = X_{B} = 0. 1 {\text{ pu}}$, $X_{E} = X_{B} = 0. 1 {\text{ pu}}$

Transmission Line: $X_{\rm BV} = 0. 3 {\text{ pu}}$, $X_{e} = X_{\rm BV} + X_{B} + X_{tE} = 0. 5 {\text{ pu}}$

Operating Condition: $V_{t} = 1. 0 {\text{ pu}}$, $P_{e} = 0. 8 {\text{ pu}}$, $V_{b} = 1. 0 {\text{ pu}}$, $f = 6 0 {\text{ Hz}}$

DC Link: $V_{\rm dc} = 2 {\text{ pu}}$, $C_{\rm dc} = 1 {\text{ pu}}$

1.5 Appendix 5. Adaptive Controller Parameters for UPFC HVDC Network

$$ \begin{aligned}& A_{m} = (q - 0. 0 1)(q - 0. 0 3)(q - 0. 0 2)(q - 0. 1)(q + 0. 1) \\ & B_{m} = q^{ 4} \\ & A_{o} = { 1} {^{\circ} B}_{m} = {^{\circ} B} = m = { 4} \\ & {^{\circ} A}_{m} = {^{\circ} A} = n = 5 \end{aligned}$$

1.6 Appendix 6. K Parameters for UPFC HVDC Network

$$\begin{aligned} & K_{1} = \frac{{\left({V_{\rm td} - I_{\rm tq} x_{d}^{\prime}} \right)(x_{dE} - x_{dt})V_{b} \sin \delta}}{{x_{d\Sigma}}} + \frac{{(x_{q} I_{\rm td} + V_{\rm tq})(x_{\rm qt} - x_{\rm qE})V_{b} \cos \delta}}{{x_{q\Sigma}}} \\ & K_{2} = \frac{{- (x_{\rm BB} + x_{E})V_{\rm td}}}{{x_{d\Sigma} x_{d}}} + \frac{{(x_{\rm BB} + x_{E})x_{d}^{\prime} I_{\rm tq}}}{{x_{d\Sigma}}} \\ & K_{3} = 1 + \frac{{(x_{d}^{\prime} - x_{d})(x_{\rm BB} + x_{E})}}{{x_{d\Sigma}}} \\ & K_{4} = \frac{{(x_{d}^{\prime} - x_{d})(x_{dE} - x_{dt})V_{b} \sin \delta}}{{x_{d\Sigma}}}\\ & K_{5} = \frac{{V_{\rm td} x_{q} (x_{\rm qt} - x_{qE})V_{b} \cos \delta}}{{V_{t} x_{q\Sigma}}} - \frac{{V_{\rm tq} x_{d}^{\prime} (x_{dtE} - x_{dt})V_{b} \sin \delta}}{{V_{t} x_{d\Sigma}}} \\ & K_{6} = \frac{{V_{\rm tq} (x_{d\Sigma} + x_{d}^{\prime} (x_{\rm BB} + x_{E}))}}{{V_{t} x_{d\Sigma}}} \\ & K_{7} = 0.25C_{\rm dc} (V_{b} \sin \delta (m_{E} \cos \delta_{E} x_{dE} - m_{B} \cos \delta_{B} x_{dt})) - \hfill \\ \frac{{m_{B} \cos \delta_{B} x_{dt}}}{{x_{d\Sigma}}} \hfill \\ V_{b} \cos \delta (m_{B} \sin \delta_{B} x_{qt} - m_{E} \sin \delta_{E} x_{\rm qe}\\ & K_{8} = - 0.25\frac{{m_{B} \cos \delta_{B} x_{\text{E}} + m_{E} \cos \delta_{E} x_{\rm BB}}}{{x_{d\Sigma}}} \\ & K_{9} = 0.25C_{\rm dc}(\frac{{m_{B} \sin \delta_{B} (m_{B} \cos \delta_{B} x_{dt} - m_{E} \cos \delta_{E} x_{dE})}}{{2x_{d\Sigma}}} + \hfill \\ \frac{{m_{E} \sin \delta_{E} (m_{E} \cos \delta_{E} x_{Bd} - m_{B} \cos \delta_{B} x_{dt})}}{{2x_{d\Sigma}}} \hfill \\ \frac{{m_{B} \cos \delta_{B} (m_{B} \sin \delta_{B} x_{qt} - m_{E} \sin \delta_{E} x_{qE})}}{{2x_{q\Sigma}}} + \hfill \\ \frac{{m_{E} \cos \delta_{E} (- m_{B} \sin \delta_{B} x_{qE} - m_{E} \sin \delta_{E} x_{Bq})}}{{2x_{q\Sigma}}}\\ & K_{\rm pe} = \frac{{(V_{\rm td} - I_{\rm tq} x_{d}^{\prime})(x_{Bd} - x_{dE})V_{\rm dc} \sin \delta_{E}}}{{2x_{d\Sigma}}} + \hfill \\ \frac{{(x_{q} I_{\rm td} + V_{\rm tq})(x_{Bq} - x_{qE})V_{\rm dc} \cos \delta_{E}}}{{2x_{q\Sigma}}}\\ & K_{p\delta E} = \frac{{\left({V_{\rm td} - I_{\rm tq} x_{d}^{\prime}} \right)(x_{Bd} - x_{dE})V_{\rm dc} m_{E} \cos \delta_{E}}}{{2x_{d\Sigma}}} + \hfill \\ \frac{{(x_{q} I_{\rm td} + V_{\rm tq})(- x_{Bq} + x_{qE})V_{\rm dc} m_{E} \sin \delta_{E}}}{{2x_{q\Sigma}}}\\ & K_{\rm pb} = \frac{{\left({V_{\rm td} - I_{\rm tq} x_{d}^{\prime}} \right)(x_{dt} - x_{dE})x_{\rm dc} \sin \delta_{B}}}{{2x_{d\Sigma}}} + \hfill \\ \frac{{(x_{q} I_{\rm td} + V_{\rm tq})(x_{\rm qt} - x_{qE})V_{\rm dc} \cos \delta_{B}}}{{2x_{q\Sigma}}}\\ & K_{p\delta B} = \frac{{\left({V_{\rm td} - I_{\rm tq} x_{d}^{\prime}} \right)(x_{dE} + x_{\rm dt})V_{\rm dc} m_{B} \cos \delta_{B}}}{{2x_{d\Sigma}}} + \hfill \\ \frac{{(x_{q} I_{\rm td} + V_{\rm tq})(- x_{\rm qt} + x_{\rm qE})V_{\rm dc} m_{B} \sin \delta_{B}}}{{2x_{q\Sigma}}}\\ & K_{\rm pd} = (V_{\rm td} - I_{\rm tq} x_{d}^{\prime})(\frac{{(x_{dt} - x_{dE})m_{B} \sin \delta_{B}}}{{2x_{d\Sigma}}} + \hfill \\ \frac{{(x_{Bd} - x_{dE})m_{E} \sin \delta_{E}}}{{2x_{d\Sigma}}} + \hfill \\ (x_{q} I_{\rm td} + V_{\rm td})(\frac{{(x_{\rm qt} - x_{qE})m_{B} \cos \delta_{B}}}{{2x_{q\Sigma}}} + \hfill \\ \frac{{(x_{Bq} - x_{qE})m_{E} \cos \delta_{E}}}{{2x_{q\Sigma}}}\\ & K_{\rm qe} = - \frac{{\left({x_{d}^{\prime} - x_{d}} \right)(x_{Bd} - x_{dE})V_{\rm dc} \sin \delta_{E}}}{{2x_{d\Sigma}}}\\ & K_{q\delta e} = - \frac{{\left({x_{d}^{\prime} - x_{d}} \right)(x_{Bd} - x_{dE})m_{E} V_{\rm dc} \cos \delta_{E}}}{{2x_{d\Sigma}}}\\ & K_{\rm qb} = - \frac{{\left({x_{d}^{\prime} - x_{d}} \right)(x_{dt} - x_{dE})V_{\rm dc} \sin \delta_{B}}}{{2x_{d\Sigma}}}\\ & K_{q\delta B} = - \frac{{\left({x_{d}^{\prime} - x_{d}} \right)(x_{dE} - x_{dt})m_{B} V_{\rm dc} \cos \delta_{B}}}{{2x_{\Sigma}}}\\ & K_{\rm qe} = - \left({x_{d}^{\prime} - x_{d}} \right)(\frac{{(x_{Bd} - x_{dE})m_{E} \sin \delta_{E}}}{{2x_{d\Sigma}}} + \frac{{(x_{dt} - x_{dE})m_{B} \sin \delta_{B}}}{{2x_{d\Sigma}}}) \\ & K_{\rm ve} = \frac{{V_{\rm td} (x_{Bq} - x_{qE})V_{\rm dc} \cos \delta_{E}}}{{2V_{t} x_{q\Sigma}}} - \frac{{V_{tq} (x_{Bd} - x_{dE})V_{\rm dc} \sin \delta_{E}}}{{2V_{t} x_{d\Sigma}}}\\ & K_{v\delta E} = \frac{{V_{\rm td} x_{q} (x_{qE} - x_{Bq})m_{E} V_{\rm dc} \sin \delta_{E}}}{{2V_{t} x_{q\Sigma}}} - \frac{{V_{\rm tq} x_{d}^{\prime} (x_{Bd} - x_{dE})m_{E} V_{\rm dc} \cos \delta_{E}}}{{2V_{t} x_{q\Sigma}}}\\ & K_{\rm vb} = \frac{{V_{td} x_{q} (x_{\rm qt} - x_{qE})V_{\rm dc} \cos \delta_{E}}}{{2V_{t} x_{q\Sigma}}} - \frac{{V_{\rm tq} x_{d}^{\prime} (x_{dt} - x_{dE})V_{\rm dc} \sin \delta_{E}}}{{2V_{t} x_{d\Sigma}}} \\ & K_{v\delta b} = \frac{{V_{\rm td} x_{q} (x_{qE} - x_{\rm qt})m_{B} V_{\rm dc} \sin \delta_{E}}}{{2V_{t} x_{q\Sigma}}} + \frac{{V_{\rm tq} m_{B} x_{d}^{\prime} (x_{dE} + x_{dt})V_{\rm dc} \cos \delta_{E}}}{{2V_{t} x_{d\Sigma}}} \\ & K_{\rm vd} = \frac{{V_{\rm td} x_{q} (x_{Bq} - x_{qE})m_{E} \cos \delta_{E}}}{{2V_{t} x_{q\Sigma}}} + \frac{{(x_{\rm qt} - x_{qE})m_{B} \cos \delta_{B}}}{{2x_{q\Sigma}}} \hfill \\ \frac{{V_{tq} m_{E} x_{d}^{\prime} (x_{Bd} - x_{dE})\sin \delta_{E}}}{{2V_{t} x_{q\Sigma}}} + \frac{{m_{B} (x_{dt} - x_{qE})\sin \delta_{E}}}{{2x_{d\Sigma}}}\\ & K_{\rm ce} = 0.25C_{\rm dc} \frac{{V_{\rm dc} \sin \delta_{E} (m_{E} \cos \delta_{E} x_{Bd} - m_{B} \cos \delta_{B} x_{dE})}}{{2x_{d\Sigma}}} + \hfill \\ \frac{{V_{\rm dc} \cos \delta_{E} (m_{E} \sin \delta_{E} x_{Bq} - m_{B} \sin \delta_{B} x_{qE})}}{{2x_{q\Sigma}}} \\ & K_{c\delta e} = \frac{{0.25m_{E}}}{{C_{\rm dc}}}(\cos \delta_{E} I_{Eq} - \sin \delta_{E} I_{Ed}) + \hfill \\ \frac{0.25}{{C_{\rm dc}}}(m_{E} V_{\rm dc} \cos \delta_{E} \frac{{(m_{E} \cos \delta_{E} x_{Bd} - m_{B} \cos \delta_{B} x_{dE})}}{{2x_{d\Sigma}}} + \hfill \\ m_{E} V_{\rm dc} \sin \delta_{E} \frac{{(m_{B} \sin \delta_{B} x_{qE} + m_{E} \sin \delta_{E} x_{Bq})}}{{2x_{q\Sigma}}}) \\ & K_{cb} = 0.25C_{\rm dc} \frac{{V_{\rm dc} \sin \delta_{B} (- m_{E} \cos \delta_{E} x_{dE} + m_{B} \cos \delta_{B} x_{dt})}}{{2x_{d\Sigma}}} + \hfill \\ \frac{{V_{\rm dc} \cos \delta_{B} (m_{B} \sin \delta_{E} x_{qt} - m_{E} \sin \delta_{E} x_{qE})}}{{2x_{q\Sigma}}} \\ & K_{c\delta B} = \frac{{0.25m_{B}}}{{C_{\rm dc}}}(\cos \delta_{B} I_{Bq} - \sin \delta_{B} I_{Bd}) + \hfill \\ \frac{0.25}{{C_{\rm dc}}}(m_{B} V_{\rm dc} \cos \delta_{B} \frac{{(m_{E} \cos \delta_{E} x_{dE} + m_{B} \cos \delta_{B} x_{dt})}}{{2x_{d\Sigma}}} + \hfill \\ m_{B} V_{\rm dc} \sin \delta_{B} \frac{{(- m_{B} \sin \delta_{E} x_{\rm qt} + m_{E} \sin \delta_{E} x_{qE})}}{{2x_{q\Sigma}}}) \end{aligned}$$